Please refer to FIG. 1 which is a circuit diagram showing a conventional programmable gain amplifier. The programmable gain amplifier includes an operational amplifier (OP) 20, a feedback resistor Rf and a resistor Rs. An input voltage signal Vs is inputted into the operational amplifier 20 from the negative terminal via the resistor Rs. The positive input terminal of the operational amplifier 20 is grounded. On the other hand, the output terminal Vo of the operational amplifier 20 is connected to its own negative input terminal via the feedback resistor Rf. The programmable gain amplifier of FIG. 1 is a Shunt-Shunt feedback amplifier with a feedback factor β. Assuming the open-loop gain of the operational amplifier is A(s), then its loop gain of the programmable gain amplifier is A(s)·β, and the closed-loop gain of the programmable gain amplifier is
            A      f        ⁡          (      s      )        =                    A        ⁡                  (          s          )                            1        +                              A            ⁡                          (              s              )                                ·          β                      .  
Another conventional programmable gain amplifier is shown in FIG. 2 which is a differential programmable gain amplifier. Compared to the programmable gain amplifier of FIG. 1, the differential programmable gain amplifier of FIG. 2 has a higher common-mode rejection ratio (CMRR) and thus better immunity to common-mode noise. The differential programmable gain amplifier includes a differential operational amplifier 40, two resistors Rs with equal resistance, and two feedback resistors Rf with equal resistance. An input voltage signal Vs is sent to the positive input terminal of the differential operational amplifier 40 via one of the resistors Rs, and also sent to the negative input terminal of the differential operational amplifier 40 via the other resistor Rs. The negative output terminal of the differential operational amplifier 40 is connected to its own positive input terminal via one of the feedback resistors Rf, while the positive output terminal of the differential operational amplifier 40 is connected to its own negative input terminal via the other feedback resistor Rf. An output signal Vo is generated between the positive output terminal and the negative output terminal of the differential operational amplifier 40. Assuming the open-loop gain of the operational amplifier is A(s), like the amplifier of FIG. 1, the loop gain of the operational amplifier of the programmable gain amplifier is A(s)·β, and the closed-loop gain of the programmable gain amplifier is
            A      f        ⁡          (      s      )        =                    A        ⁡                  (          s          )                            1        +                              A            ⁡                          (              s              )                                ·          β                      .  
Accordingly, by varying the feedback factor β, the gain of the programmable gain amplifier can be dynamically adjusted. Generally, the feedback factor β varies by adjusting the resistance of the resistor Rf.
Please refer to FIG. 3 which is a Bode plot showing the relationship between the open-loop gain A(s) and the closed loop-gain G1, G2 of an operational amplifier. Assuming the equation
      A    ⁡          (      s      )        =            A      0                      (                  1          +                      s                          ω              1                                      )            ⁢              (                  1          +                      s                          ω              2                                      )            ⁢              (                  1          +                      s                          ω              3                                      )            applies, where Ao is a constant, ω3ω2ω1 are pole frequencies with ω3>ω2>ω1, the phase φ is expressed by φ=−[tan−1(ω/ω1)+tan−1(ω/ω2)+tan−1(ω/ω3)].
It is understood from the Bode plot of FIG. 3 that the open-loop gain A(s) is a horizontal line with a constant gain G0 at frequencies smaller than the frequency ω1. Then the gain decays with the increase of frequency. The open-loop gain A(s) becomes slantingly linear with a slop of −20 dB/decade between the frequency ω1 and the frequency ω2. Afterwards, the slope becomes −40 dB/decade with frequency increasing up to the frequency ω3 and further becomes −60 dB/decade with frequency higher than the frequency ω3. Since ω1 is a cut-off frequency, the bandwidth of the open-loop gain A(s) is equal to ω1.
When the feedback factor β is equal to β1, the Bode plot shows a closed loop gain G1 with a cut-off frequency ωx, and then decays with a slope of −20 dB/decade. Since the ωx is the cut-off frequency, the bandwidth closed loop gain G1 is equal to ωx. When the feedback factor β is equal to β2, the Bode plot shows a closed loop gain G2 with a cut-off frequency ωy, and then decays with a slope of −20 dB/decade. Since the ωy is a cut-off frequency, the bandwidth of the closed loop gain G2 is equal to ωy.
It is also understood from FIG. 3 that, in the conventional programmable gain amplifier, the gain can be adjusted by way of varying the feedback factor β, while the bandwidth of the programmable gain amplifier is also changed accordingly. In other words, as the gain of the programmable gain amplifier increases, the bandwidth of the programmable gain amplifier is reduced. As the gain of the programmable gain amplifier is reduced, the bandwidth of the programmable gain amplifier is increased. For some applications, such as a video tuner or communication applications, the bandwidth of a programmable gain amplifier is desired to be stable under a variety of gains so as to avoid undesirable serious signal deterioration of some channels with gain changes.